The protesters scattered into side streets, where they chanted anti-Morsi slogans as the Islamists shouted, "The people demand the implementation of God's law!"
After a brief lull, hundreds of Morsi opponents arrived and began throwing firebombs at the president's backers, who responded with rocks. The clashes continued well after nightfall and spread from the immediate vicinity of the palace to residential streets nearby.
The deployment of hundreds of riot police did not stop the fighting. The police later fired tear gas to disperse Morsi's opponents. Volunteers ferried the wounded on motorcycles to waiting ambulances, which rushed them to hospitals.
"I voted for Morsi to get rid of Hosni Mubarak. I now regret it," Nadia el-Shafie yelled at Brotherhood supporters on a side street.
"God is greater than you! Don't think this power or authority will add anything to you. God made this revolution, not you!" the tearful woman said as she was led away from the crowd of Islamists.
"May God protect Egypt and its president," read a banner hoisted atop a truck brought by the Islamists, as a man using a loudspeaker recited verses from the Quran.
"We came to support the president. We feel there is a legitimacy that someone is trying to rob," said Rabi Mohammed, a Brotherhood supporter. "People are rejecting democratic principles using thuggery."
The Islamists portrayed their attack on opposition protesters as defense of the revolution.
The clashes, said top Brotherhood leader Essam el-Erian, pitted "those who are protecting the legitimacy and the revolution against the counterrevolution and coup plotters."
Vice President Mahmoud Mekki called for a dialogue with the opposition to reach a consensus on disputed articles of the constitution, which he put at 15 out of a total of 234. The referendum must go ahead, he said, adding that he was acting in a personal capacity, not on behalf of Morsi.
Speaking to reporters, ElBaradei said there would be no dialogue unless Morsi rescinded his decrees and shelved the draft constitution.
Asked to comment on Mekki's offer, he said: "With all due respect, we don't deal with personal initiatives. If there is a genuine desire for dialogue, the offer must come from President Morsi."
Morsi's Nov. 22 decrees were followed last week by the constitutional panel pushing through the draft constitution without the participation of liberal and Christian members. Only four women, all Islamists, attended the session.
If the referendum goes ahead as scheduled and the draft constitution is adopted, elections for parliament's lawmaking lower chamber will be held in February.
AP reporters Maggie Michael and Sarah El Deeb contributed to this report.
Copyright 2012 The Associated Press. All rights reserved. This material may not be published, broadcast, rewritten or redistributed.